General Theory of Music by Icosahedron 2: Analysis of musical pieces by the exceptional musical icosahedra

TL;DR

This paper introduces a novel analytical framework for musical harmony using exceptional musical icosahedra, employing golden triangles and gnomons to decompose and analyze chords and compositions.

Contribution

It develops the concept of golden neighborhoods and golden decomposition, linking icosahedral symmetry with harmonic analysis, and applies this to analyze complex chords and Bach's prelude.

Findings

Dominant seventh chord is the only golden singular in certain icosahedra types.

Half-diminished seventh chord is golden singular in other icosahedra types.

Seven golden figure combinations represent all measures of Bach's prelude.

Abstract

We propose a new way of analyzing musical pieces by using the exceptional musical icosahedra where all the major/minor triads are represented by golden triangles or golden gnomons. First, we introduce a concept of the golden neighborhood that characterizes golden triangles/gnomons that neighbor a given golden triangle or gnomon. Then, we investigate a relation between the exceptional musical icosahedra and the neo-Riemannian theory, and find that the golden neighborhoods and the icosahedron symmetry relate any major/minor triad with any major/minor triad. Second, we show how the exceptional musical icosahedra are applied to analyzing harmonies constructed by four or more tones. We introduce two concepts, golden decomposition and golden singular. The golden decomposition is a decomposition of a given harmony into the minimum number of harmonies constructing the given harmony and represented by the golden figure (a golden triangle, a golden gnomon, or a golden rectangle). A harmony is golden singular if and only if the harmony does not have golden decompositions. We show results of the golden analysis (analysis by the golden decomposition) of the tertian seventh chords and the mystic chord. While the dominant seventh chord is the only tertian seventh chord that is golden singular in the type 1[star] and the type 4[star] exceptional musical icosahedron, the half-diminished seventh chord is the only tertian seventh chord that is golden singular in the type 2 [star] and the type 3[star] exceptional musical icosahedron. Last, we apply the golden analysis to the famous prelude in C major composed by Johann Sebastian Bach (BWV 846). We found 7 combinations of the golden figures on the type 2 [star] or the type 3 [star] exceptional musical icosahedron dually represent all the measures of the BWV 846.